Electrophoresis of charged polymers: Simulation and scaling in a lattice model of reptation

We report numerical results on the repton model of Rubinstein [Phys. Rev. Lett. 59, 1946 (1987)] as adapted by Duke [Phys. Rev. Lett. 62, 2877 (1989)] as a model for the gel electrophoresis of DNA. We describe an efficient algorithm with which we have simulated chains of N reptons with N several hundred in some instances. The diffusion coefficient D in the absence of an external electric field is obtained for N≤100; we find ${\mathit{N}}^2$D=1/3(1+5${\mathit{N}}^{\mathrm{-}2/3}$) for large N. The coefficient 1/3 is in accord with the analytical results of Rubinstein and of van Leeuwen and Kooiman [Physica A 184, 79 (1992)]. The drift velocity v in a static field of variable strength E is obtained for various N and NE up to N=30 when NE is as small as 0.01 and up to N=400 when NE is as large as 20. We find that v has a finite, nonzero limit as N→∞ at fixed E and that this limit is proportional to ‖E‖E, in accord with the conclusions of Duke, Semenov, and Viovy [Phys. Rev. Lett. 69, 3260 (1992)] for a different but related model. In a scaling limit in which N→∞ and E→0 the drift velocity in the repton model is fitted well by the formula ${\mathit{N}}^2$v=NE[(1/3${)}^2$+(2NE/5${)}^2$ ${]}^{1/2}$ for all values of the scaling variable NE. We present a scaling analysis complementary to that of Duke, Semenov, and Viovy with which we rationalize the ‖E‖E behavior of the limiting drift velocity.